To prove (F\(_i\) ) and (H\(_i\) ), we will need some bounds from [1]}. The bounds in [1]} apply to \(\psi _i\)  rather than \(f_i\) , but for each \(v\in U\) , there is an \(i^{\prime }\le i\)  such that \(f_i\)  and \(\psi _{i^{\prime }}\)  agree on a neighborhood of \(v\) . Therefore, by Lemma 3.10 of [1]},
\(\left\Vert \frac{\partial f_{i}}{\partial z}\right\Vert _{\infty } \le \max _{i^{\prime }\le i} \left\Vert \frac{\partial \psi _{i^{\prime }}}{\partial z}\right\Vert _\infty \le \max _{i^{\prime }\le i} 2\rho ^{i^{\prime }-1} \le 2\rho ^{i-1},\) 
